1. Field of the Invention
The present invention relates to a position detection method and position detector, an exposure method and exposure apparatus, and a device and device manufacturing method and, more particularly, to a position detection method and position detector for obtaining arrangement information of divided areas on an object, an exposure method and exposure apparatus using the position detection method, and a device manufactured using the exposure method and a manufacturing method thereof.
2. Description of the Related Art
In a lithography process for making semiconductor devices, liquid crystal display devices, and the like, an exposure apparatus which transfers a pattern formed on a mask or reticle (to be generally referred to as a xe2x80x9creticlexe2x80x9d hereinafter) onto a substrate (to be referred to as a xe2x80x9csensitive substrate or waferxe2x80x9d as needed hereinafter) such as a wafer, glass plate, or the like coated with a resist or the like via a projection optical system is used. As such an exposure apparatus, stationary exposure type projection exposure apparatus such as a so-called stepper, or a scanning exposure type projection exposure apparatus such as a so-called scanning stepper is mainly used.
In these exposure apparatuses, position adjustment (alignment) between a reticle and wafer must be accurately performed prior to exposure. To achieve this alignment, position detection marks (alignment marks) are formed (transferred by exposure) in the previous lithography process on the wafer in each of shot regions, and the position of the wafer (or a circuit pattern on the wafer) can be detected by detecting the positions of the alignment marks. Alignment is performed on the basis of the detection results of the positions of the wafer (or a circuit pattern on the wafer).
Such alignment schemes include a die-by-die scheme for performing alignment by detecting alignment marks in each shot, and an enhanced global alignment (to be abbreviated as xe2x80x9cEGAxe2x80x9d hereinafter) scheme for, after measuring alignment marks (position adjustment marks transferred together with a circuit pattern) at several positions in a wafer, computing arrangement coordinate positions of each shot area by a statistical scheme such as a least square approximation and, upon exposure, stepping a wafer using the accuracy of a wafer stage and the computation result. This EGA scheme is disclosed in, e.g., Japanese Patent Laid-Open No. 61-44429 and corresponding U.S. Pat. No. 4,780,617. Of these schemes, the EGA scheme is prevalently used nowadays in terms of the throughput of the apparatus.
In this EGA scheme, in order to determine a plurality of parameters that uniquely specifies the actual arrangement coordinate positions of shot areas relative to ideal arrangement coordinate positions in terms of design, more positions of alignment marks than the minimum number of those required to obtain the plurality of parameters are measured. Then, statistically valid parameter values are determined using a statistical scheme such as a least square approximation.
Upon applying such statistical scheme, error analysis is performed under the premise that xe2x80x9call position measurement results of alignment marks have the same reliabilityxe2x80x9d (hereafter, this case is referred to as xe2x80x9cprior art 1xe2x80x9d).
Also, as disclosed in Japanese Patent Publication No. 7-120621, a technique for determining the predetermined parameters by executing a fuzzy process based on fuzzy inference using statistical values such as the average value and variance of the measured positions of the alignment marks and obtaining the arrangement coordinate positions of the shot areas has also been proposed (hereafter, this case is referred to as xe2x80x9cprior art 2xe2x80x9d).
When all alignment marks are equally formed, the premise of prior art 1 that xe2x80x9call position measurement results of alignment marks have the same reliabilityxe2x80x9d is true, but not true when the shapes of alignment marks differ depending on their positions on a substrate. Therefore, when the shapes of alignment marks differ depending on their positions on a substrate, all position measurement results, whether their reliabilities are high or low, equally contribute to determination of the arrangement coordinate positions of shot areas their different reliabilities.
The accuracy of the arrangement coordinate position determination of shot areas determined under the above premise suffices to achieve conventionally required exposure accuracy, but does not suffice for the increase of integration degree in recent years.
Prior art 2 is free from any problem of the accuracy of the arrangement coordinate position determination of shot areas unlike in prior art 1. However, since prior art 2 requires a huge computation volume for fuzzy inference, a long period of time is required to determine the arrangement coordinate positions of shot regions, and this makes it difficult to improve the throughput of exposure. In order to prevent such low throughput, a large-scale computation resource is needed. However, the use thereof causes the whole exposure apparatus to be large and complicated.
Upon applying the conventional statistical scheme, the positions of alignment marks to be measured on a wafer are determined empirically or on a trial-and-error basis that after transferring a pattern onto a wafer while aligning the wafer using a temporarily selected sample set, the same patterns on the wafer are measured, and that if desired results are not obtained, another sample set is selected.
As described above, in the conventional method, a sample set of alignment marks as a subset of a set of all alignment marks is determined by an aleatory method, and the validity of determination of that sample set is not quantitatively evaluated. Therefore, it is not guaranteed that the error distribution of the positions of a plurality of alignment marks as elements of a sample set determined by the conventional method appropriately reflects the error distribution of positions of all alignment marks.
For trying to solve this problem, there is the following method: the position control using a plurality of parameters which are obtained using a provisional sample set determined empirically or arbitrarily and uniquely specify the arrangement coordinate positions of shot regions, when the sample set includes shot areas (so-called xe2x80x9cisolated shotsxe2x80x9d) having much larger alignment errors than those of other shot regions, alignment marks contained in such isolated shots are excluded from the sample set. This method presupposes that only few isolated shots exist, and that those alignment marks cause the decrease of alignment accuracy for all shot regions.
However, if alignment marks of two isolated shot regions, of which pattern shift directions are almost opposite to each other (i.e., the two shot areas having negative correlation), are selected, high-accuracy alignment is possible. Therefore, exclusion of the measured position information of alignment marks contained in an isolated shot area may result in an alignment accuracy decrease.
In the case where a wafer is aligned on the basis of the position measurement results of alignment marks included in a subset (sample set) selected from a large number of alignment marks, upon examining the validity of a method for selecting the desired sample set, evaluating separately individual alignment marks in the sample set does not have much sense. This is because it is ideal that the sample set broadly reflects the entire set and because it is preferable that alignment marks in the sample set preferably have a position distribution that corresponds to that of alignment marks in the entire set. For example, in the case where one of five alignment marks in a sample set is an alignment mark contained in an isolated shot region, if one fifth of all the alignment marks are contained in isolated shot regions, the sample set is more valid than a sample set excluding alignment marks of isolated shot regions. That is, position errors of measured alignment marks reflect the position distribution of all the alignment marks somehow, and should not be carelessly ignored. However, there have been no proposals concerning a method for selecting a sample set from the entire set of alignment marks formed on a substrate so as to perform statistically valid position control and alignment on the basis of the position measurement results of alignment marks in the sample set.
Furthermore, although it is not appropriate to exclude alignment marks in descending order of position error amounts in order to reduce the number of sample alignment marks and alignment measurement time, there have been no proposals concerning a method for reducing the number of sample alignment marks while maintaining alignment accuracy.
That is, an alignment technology that can meet recent requirements of improved exposure accuracy and throughput is needed.
The present invention has been made considering the above situation, and a first object of the present invention is to provide a position detection method and position detector which can accurately and efficiently detect arrangement information of divided areas on an object.
A second object of the present invention is to provide an exposure method and exposure apparatus that can transfer a predetermined pattern onto a substrate with high accuracy.
A third object of the present invention is to provide a device on which fine patterns are accurately formed.
A fourth object of the present invention is to provide a manufacturing method of manufacturing a device on which fine patterns are accurately formed.
According to the first aspect, there is provided a first position detection method for detecting position information of any area on an object provided with a plurality of position-measurement-points, the position detection method comprising a measurement step of selecting more position-measurement-points than a minimum number of measurements required to calculate values of a predetermined number of parameters, which uniquely specify position information of any area on the object, from the plurality of position-measurement-points and measuring pieces of position information of the respective selected position-measurement-points; an estimation step of calculating respective positions of the selected position-measurement-points, based on the measurement results of the pieces of position information, and estimating probability density functions which each represent occurrence probability of the calculated position for respective one of the selected position-measurement-points; a probability density calculation step of calculating probability density of the calculated position of each of the position-measurement-points, based on respective one of the probability density functions; and a parameter calculation step of evaluating an error of each calculated position relative to respective reference position while using the respective calculated probability density""s value as a piece of weight information and calculating values of the predetermined number of parameters, based on the evaluated errors.
According to this method, pieces of position information of selected position-measurement-points are measured, and the positions and probability densities of the selected position-measurement-points are calculated on the basis of the measurement results. Upon calculating the statistically most valid values of a predetermined number of parameters, which uniquely specify position information of any area on an object, while using the calculated probability densities as pieces of information each representing the certainty of the position of a respective position-measurement-point, the error between each calculated position and a respective reference position is weighted in accordance with the certainty of the calculated position of the respective position-measurement-point, i.e., the probability density at the calculated position of the respective position-measurement-point. That is, if the probability density is large, the weight is large, and if the probability density is small, the weight is small. As a result, when the calculated position of a position-measurement-point has high certainty, the degree of influence of the error of the calculated position of the position-measurement-point relative to its reference position is high; when the calculated position of a position-measurement-point has low certainty, the degree of influence of the error of the calculated position of the position-measurement-point relative to its reference position is low. Therefore, since statistically valid values of a predetermined number of parameters which uniquely specify position information of any area on an object can be calculated while rationally reflecting respective certainties of the calculated positions of position-measurement-points, the position of a area of interest on the object can be accurately detected.
In the first position detection method of the present invention, the reference positions can be determined in advance on the basis of design information.
The probability density of each calculated mark (a position-measurement-point) position directly reflects the certainty of a respective mark position. Therefore, in the first position detection method of the present invention, the errors are evaluated by multiplying the errors of the calculated positions relative to the respective reference positions by the respective probability densities of the calculated positions.
In the first position detection method of the present invention, normal distributions can be adopted as the probability density functions. In this way, it is particularly valid to presume the occurrence probability distributions to be a normal distribution when variations of errors of the calculated mark positions relative to the respective reference positions are expected to be random like normal random numbers. When the occurrence probability distribution is known, a probability density function of that probability distribution can be used instead of a normal distribution. On the other hand, when the occurrence probability distribution is unknown, it is rational to presume the occurrence probability distribution to be a normal distribution, which is the most general probability distribution.
In the first position detection method of the present invention, position measurement marks can be formed at the position-measurement-points. In such case, the position of each position-measurement-point can be measured by detecting a respective position measurement mark. Note that the position measurement mark can be, e.g., a line-and-space mark, box-in-box mark, and the like.
In this case, a plurality of position detection marks formed at the plurality of position-measurement-points can include a first number of first marks, of which surface states change in a first direction, and the position information of each first mark measured in the measurement step can be position information of a plurality of feature portions in the first direction of each first mark. In such a case, the position of a first mark in the first direction can be calculated by measuring and processing the position information of the first mark. Also, a probability density function that represents the occurrence probability of the calculated position can be estimated based on its design reference position and measured position information. When the first mark periodically changes in the first direction like a line-and-space mark, the average value of positions of a plurality of feature portions in the first direction such as the boundaries between lines and spaces, which represents the central position of the first mark, can be used as the position of the first mark, and the probability density function that represents the occurrence probability of the central position can be estimated.
The position information of each selected first mark can be measured in the measurement step at a plurality of positions in a direction perpendicular to the first direction. In this case, since the number of pieces of position information to be processed increases, the position of the first mark in the first direction can be accurately calculated, and the probability density function that represents the occurrence probability of the calculated position can be accurately estimated.
The surface state of each first mark can also change in a second direction different from the first direction, and the position information of the first mark measured in the measurement step can include position information, in the first direction, of a plurality of feature portions lined in the first direction of the first mark, and position information, in the second direction, of a plurality of feature portions lined in the second direction of the first mark. In this case, the two-dimensional position of the first mark can be calculated based on the measurement result of the position information of the first mark, and the probability density function that represents the occurrence probability of the calculated position can be estimated based on the design reference position and measured position information. That is, information that pertains to the two-dimensional position of the object can be calculated.
In the measurement step, for each selected first mark, at least one of position information in the first direction of a plurality of feature portions in the first direction and position information in the second direction of a plurality of feature portions in the second direction can be measured. In this case, since the number of pieces of position information to be processed increases for at least one of the first and second directions, the position of the first mark in a direction in which the number of pieces of position information to be processes has increased can be accurately calculated, and the probability density function that represents the occurrence probability of the calculated position can be accurately estimated.
The plurality of marks can further include a second number of second marks, of which surface states change in a second direction different from the first direction, and the position information of each second mark measured in the measurement step can be position information of a plurality of feature portions in the second direction of the second mark. In this case, the position of the second mark in the second direction can be calculated by measuring and processing position information of the second mark in the same manner as for the first mark, and a probability density function that represents the occurrence probability of the calculated position can be estimated based on its design reference position and measured position information. That is, information that pertains to a two-dimensional position of the object can be calculated.
In the measurement step, the position information of each selected second mark can be measured at a plurality of positions in a direction perpendicular to the second direction. In this case, since the number of pieces of position information to be processed increases, the position of the second mark in the second direction can be accurately calculated, and the probability density function that represents the occurrence probability of the calculated position can be accurately estimated.
Furthermore, in the first position detection method of the present invention in which position measurement marks are formed at position-measurement-points, a plurality of divided areas can be arranged on an object, and position measurement marks can be contained in each of the plurality of divided areas. In this case, the arrangement coordinate position of each divided area on the object can be accurately detected.
In addition, the predetermined number of parameters can include parameters associated with representative points of the plurality of divided areas. In this case, the arrangement of the representative points, e.g. the central points, of the plurality of divided areas on the object can be calculated, the arrangement being referred to as an arrangement coordinate system.
Note that the predetermined number of parameters can further include parameters associated with points other than the representative points of the plurality of divided areas. In this case, in addition to the arrangement coordinate system of representative points of the plurality of divided areas on the object, a divided area coordinate system that specifies the direction of pattern transfer, scale, and the like on the divided areas can be calculated.
According to the second aspect, there is provided a second position detection method for detecting position information of any area on an object provided with a first number of position-measurement-points, the position detection method comprising a first step of selecting a plurality of measurement point subsets which each consist of a third number of position-measurement-points and are different from one another, the third number being larger than a second number and smaller than the first number, the second number being a minimum number of measurement points required to calculate a predetermined number of parameters that uniquely specify position information of any area on the object; and a second step of statistically calculating, for each of the plurality of measurement point subsets, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of the third number of position-measurement-points. For each measurement point subset, the certainty of the estimations of the predetermined number of parameters is calculated using the calculated estimations, and is determined in accordance with position errors of position-measurement-points, which are used to calculate the estimations, relative to respective expected positions. If the deviation of position errors of the position-measurement-points used to calculate the estimations is large, the certainty is low; if the deviation of the position errors is small, the certainty is high.
According to this method, a plurality of different measurement point subsets are selected, and for each of the measurement point subsets, estimations and their certainty of the predetermined number of parameters which uniquely specify position information of any area on an object are statistically calculated. The estimations and their certainty of the predetermined number of parameters (to be also referred to as xe2x80x9cposition parametersxe2x80x9d hereinafter) of each measurement point subset reflect the position distribution of all position-measurement-points. Therefore, the position distribution of all position-measurement-points can be accurately estimated based on respective groups of the estimations and their certainty of the predetermined number of parameters for the plurality of measurement point subsets that are selected empirically or arbitrarily.
The second position detection method can further comprise the third step of obtaining statistically valid values of the predetermined number of parameters, based on the respective groups of estimations and certainty for the plurality of measurement point subsets calculated in the second step. In this case, since statistically valid position parameter values are calculated on the basis of respective groups of the estimations and their certainty of position parameters for the measurement point subsets, and the groups of the estimations and certainty each statistically reflect the predetermined number of parameters that are statistically determined by broadly sampling from all position-measurement-points, statistically valid values of the predetermined number of parameters for all the position-measurement-points can be accurately calculated.
Furthermore, the statistically valid value of each of the predetermined number of parameters is obtained by calculating average of the corresponding estimations weighted with respective certainties, each of the certainties representing a piece of weight information for the respective estimation. In this case, since the weighted mean of estimations is calculated using the respective certainties of the estimations as respective weights of the estimations, the rational evaluation of the estimations can be performed in which estimations with a low certainty contribute less, and in which other estimations with a high certainty contribute more. And statistically valid values of the predetermined number of parameters for all the position-measurement-points can be accurately and easily calculated.
In the second position detection method of the present invention, in the second step, certainties of position measurement results of the position-measurement-points can be taken in account for calculating the estimations and certainty thereof. In this case, since the estimations and their certainty of the predetermined number of parameters are calculated considering the certainties of the position measurement results at the position-measurement-points, statistically more valid values of the predetermined number of parameters can be calculated.
In addition, the second step can comprise an estimation step of calculating, for each of the plurality of measurement point subsets, respective positions of the third number of position-measurement-points based on measurement results of the third number of position-measurement-points and estimating probability density functions that each represent occurrence probability of the calculated position of the respective, selected position-measurement-point; a probability density calculation step of calculating respective probability densities of the calculated positions of the position-measurement-points, based on the probability density functions; and a parameter calculation step of evaluating an error of each of the calculated positions relative to respective reference position using the respective calculated probability density""s value as a piece of weight information and calculating estimations of the predetermined number of parameters, based on the evaluated errors. In this case, since errors between the calculated positions and respective reference positions are weighted in accordance with the information of the certainties of the calculated positions of the position-measurement-points, i.e. the probability densities at the calculated positions of the position-measurement-points, statistically valid estimations of the predetermined number of parameters can be calculated which uniquely specify position information of any area on an object and rationally reflect the certainties of the calculated positions of the position-measurement-points.
In the second position detection method of the present invention, position measurement marks can be formed at the position-measurement-points as in the first position detection method of the present invention. And, a plurality of divided areas can be arranged on the object, and position measurement marks can be contained in each of the plurality of divided areas.
According to the third aspect, there is provided a third position detection method for detecting position information of any area on an object provided with a first number of position-measurement-points, the position detection method comprising a first step of selecting a first measurement point subset which consists of a third number of position-measurement-points, the third number being larger than a second number and smaller than the first number, the second number being a minimum number of measurement points required to calculate a predetermined number of parameters that uniquely specify position information of any area on the object; a second step of selecting a plurality of second measurement point subsets which each consist of a fourth number of position-measurement-points and are different from one another, the fourth number being larger than the second number and smaller than the third number; and a third step of statistically evaluating possibility of replacing the first measurement point subset by one of the plurality of second measurement point subsets, based on measurement results of the third number of position-measurement-points composing the first measurement point subset and measurement results of sets of the fourth number of position-measurement-points each composing one of the second measurement point subsets, the first measurement point subset being used to calculate the predetermined number of parameters.
According to this, for each of the plurality of second measurement point subsets, it is evaluated whether or not it is possible to replace the initially selected sample set (first measurement point subset) by a sample set including a smaller number of elements. That is, to determine whether or not it is possible to reduce the number of position-measurement-points used to calculate the predetermined number of parameters, it is evaluated, based on the position measurement results at the position-measurement-points of the first measurement point subset and those of each second measurement point subset, whether or not the position error distribution of position-measurement-points in the second measurement point subset is similar to that of position-measurement-points of the first measurement point subset, in other words, whether or not the second measurement point subset and the first measurement point subset equally reflect the entire set of all position-measurement-points. Therefore, upon reducing the number of position-measurement-points as elements of a sample set, statistical validity of the calculated values of the predetermined number of parameters can be maintained.
In the third position detection method of the present invention, the third step can comprise a fourth step of statistically calculating estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of the third number of position-measurement-points composing the first measurement point subset; a fifth step of statistically calculating estimations of the predetermined number of parameters and certainty of the estimations for each of the plurality of second measurement point subsets, based on measurement results of the fourth number of position-measurement-points; and a sixth step of comparing the estimations and certainty of the first measurement point subset with the estimations and certainty for each of the plurality of second measurement point subsets and evaluating possibility of replacing the first measurement point subset by one of the plurality of second measurement point subsets, the first measurement point subset being used to calculate the predetermined number of parameters.
In this case, the estimations and their certainty of the predetermined number of parameters calculated based on the position measurement results at the position-measurement-points of the first measurement point subset are compared with those calculated based on the position measurement results at the position-measurement-points of each second measurement point subset. In this comparison, the certainties of respective groups of the estimations of the two measurement point subsets are compared as well as the groups of the estimations, the certainties each reflecting deviation of the position error distribution of position-measurement-points of the respective measurement point subset. And by examining the two comparison results, the position error distribution of position-measurement-points of the first measurement point subset is compared with that of position-measurement-points of the second measurement point subset. Therefore, it can be determined whether or not one of the plurality of second measurement point subsets and the first measurement point subset equally reflect the entire set of all position-measurement-points.
Furthermore, in the fourth step, certainties of position measurement results of the position-measurement-points can be taken in account for calculating the estimations and certainty thereof. In this case, since the estimations and their certainty of the predetermined number of parameters are calculated considering the certainties of the position measurement results at the position-measurement-points, statistically more valid values of the predetermined number of parameters can be calculated.
Additionally, the fourth step can comprise an estimation step of calculating respective positions of the third number of position-measurement-points, which compose the first measurement point subset, based on measurement results of the third number of position-measurement-points and estimating probability density functions that each represent occurrence probability of the calculated position of a respective point of the third number of position-measurement-points; a probability density calculation step of calculating respective probability densities of the calculated positions of the position-measurement-points, based on the probability density functions; and a parameter calculation step of evaluating an error of each of the calculated positions relative to respective reference position using the respective calculated probability density""s value as a piece of weight information and calculating estimations of the predetermined number of parameters, based on the evaluated errors. In this case, since, for the third number of position-measurement-points composing the first measurement point subset, errors between the calculated positions and their reference positions are weighted in accordance with the information of the certainties of the calculated positions of the position-measurement-points, i.e. the probability densities at the calculated positions of the position-measurement-points, statistically valid estimations of the predetermined number of parameters can be calculated which uniquely specify position information of any area on an object and rationally reflect the certainties of the calculated positions of the position-measurement-points.
In addition, in the fifth step, certainties of position measurement results of the position-measurement-points can be taken into account upon calculating the estimations and certainty thereof. Moreover, the fifth step can comprise an estimation step of calculating respective positions of the fourth number of position-measurement-points, for each of the plurality of second measurement point subsets, based on measurement results of the fourth number of position-measurement-points and estimating probability density functions that each represent occurrence probability of the calculated position of a respective point of the fourth number of position-measurement-points; a probability density calculation step of calculating respective probability densities of the calculated positions of the position-measurement-points, based on the probability density functions; and a parameter calculation steep of evaluating an error of each of the calculated positions relative to respective reference position using the respective calculated probability density""s value as a piece of weight information and calculating estimations of the predetermined number of parameters, based on the evaluated errors.
In addition, in the third position detection method according to this invention, the third step can comprise a fourth step of statistically calculating, for each of the second measurement point subsets, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of the fourth number of position-measurement-points; a fifth step of statistically calculating position errors of the position-measurement-points of the first measurement point subset through use of the estimations of the predetermined number of parameters calculated in the fourth step and evaluating possibility of replacing the first measurement point subset by one of the plurality of second measurement point subsets.
In this case, by calculating position errors of position-measurement-points composing the first measurement point subset by using the estimations, of the predetermined number of parameters, calculated on the basis of the position measurement results at the position-measurement-points of each second measurement point subset, the position error distribution of position-measurement-points in the first measurement point subset can be obtained. Therefore, without calculating the estimations and their certainty of the predetermined number of parameters on the basis of the position measurement results at the position-measurement-points of the first measurement point subset, it can be determined whether or not one of the plurality of second measurement point subsets and the first measurement point subset equally reflect the entire set of all position-measurement-points.
Especially, in the case where a second measurement point subset to be compared is a subset of the first measurement point subset, by calculating position errors of position-measurement-points which are included in the first measurement point subset but not included in the second measurement point subset, the estimations and their certainty of the predetermined number of parameters can be calculated for the case where the first measurement point subset is used as a sample set. Therefore, it can be quickly determined whether or not one of the plurality of second measurement point subsets and the first measurement point subset equally reflect the entire set of all position-measurement-points.
Furthermore, in the fourth step, certainties of position measurement results of the position-measurement-points can be taken in account for calculating the estimations and certainty thereof. In this case, since the estimations and their certainty of the predetermined number of parameters are calculated considering the certainties of the position measurement results at the position-measurement-points, statistically more valid values of the predetermined number of parameters can be calculated.
In addition, the fourth step can comprise an estimation step of calculating respective positions of the fourth number of position-measurement-points, for each of the plurality of second measurement point subsets, based on measurement results of the fourth number of position-measurement-points and estimating probability density functions that each represent occurrence probability of the calculated position of a respective point of the fourth number of position-measurement-points; a probability density calculation step of calculating respective probability densities of the calculated positions of the position-measurement-points, based on the probability density functions; and a parameter calculation step of evaluating an error of each of the calculated positions relative to respective reference position using the respective calculated probability density""s value as a piece of weight information and calculating estimations of the predetermined number of parameters, based on the evaluated errors. In this case, since, for each of the second measurement point subsets, errors between the calculated positions and their reference positions are weighted in accordance with the information of the certainties of the calculated positions of the position-measurement-points, i.e. the probability densities at the calculated positions of the position-measurement-points, statistically valid estimations of the predetermined number of parameters can be calculated which uniquely specify position information of any area on an object and rationally reflect the certainties of the calculated positions of the position-measurement-points.
Furthermore, the third position detection method according to this invention can further comprise the fourth step which, if the third step finds second measurement point subsets that can replace the first measurement point subset, selects the most valid, second measurement point subset for replacement and adopts estimations of the predetermined number of parameters, calculated based on measurement results of the fourth number of position-measurement-points of the selected second measurement point subset, as values thereof, and which, if the third step finds no second measurement point subsets that can replace the first measurement point subset, adopts estimations of the predetermined number of parameters, calculated based on measurement results of the second number of position-measurement-points of the first measurement point subset, as values thereof.
In this case, if the third step finds second measurement point subsets that can replace the first measurement point subset, i.e. if the number of position-measurement-points can be reduced maintaining the statistical validity, the most valid, second measurement point subset for replacement is adopted as the sample set. On the other hand, if the third step finds no second measurement point subsets that can replace the first measurement point subset, i.e. if the number of position-measurement-points can not be reduced maintaining the statistical validity, the first measurement point subset is adopted as the sample set. Then, the estimations of the predetermined number of parameters calculated based on the position measurement results at the position-measurement-points of the sample set are adopted as the values of the predetermined number of parameters. Therefore, the number of position-measurement-points used to calculate the values of the predetermined number of parameters can be reduced while maintaining the statistical validity, and improvement of the position detection speed can be achieved maintaining the accuracy.
In the third position detection method according to this invention, position measurement marks can be formed at the position-measurement-points in the same manner as in the first position detection method, and a plurality of divided areas, each of which is provided with the position measurement marks, can be arranged on the object.
According to the fourth aspect of this invention, there is provided a fourth position detection method for detecting position information of any area on an object provided with a first number of position-measurement-points, the position detection method comprising a first step of selecting a plurality of first measurement point subsets which each consist of a third number of position-measurement-points and are different from one another, the third number being larger than a second number and smaller than the first number, the second number being a minimum number of measurement points required to calculate a predetermined number of parameters that uniquely specify position information of any area on the object; a second step of selecting a plurality of second measurement point subsets which each consist of a fourth number of position-measurement-points and are different from one another, the fourth number being larger than the second number and smaller than the third number; and a third step of statistically evaluating possibility of replacing the plurality of first measurement point subsets by one of the plurality of second measurement point subsets, as a measurement point set used to calculate the predetermined number of parameters.
According to this method, for each of the plurality of second measurement point subsets, it is evaluated whether or not it is possible to replace the plurality of initially selected sample sets (the plurality of first measurement point subsets) by one sample set composed of a fewer number of elements. That is, to determine whether or not the number of position-measurement-points used to calculate values of the predetermined number of parameters and the processing volume of the position measurement results can be reduced, it is evaluated whether or not the position error distribution of position-measurement-points in one of the plurality of second measurement point subsets is similar to a position error distribution, for all position-measurement-points, estimated based on the position measurement results at the position-measurement-points of the plurality of first measurement point subsets. Therefore, upon reducing the number of position-measurement-points as elements of a sample set and reducing the processing volume of the position measurement results, statistical validity of the calculated values of the predetermined number of parameters can be maintained.
In the fourth position detection method according to this invention, the third step can comprise a fourth step of statistically calculating, for each of the plurality of first measurement point subsets, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of the third number of position-measurement-points; a fifth step of calculating statistically valid estimations of the predetermined number of parameters and certainty of the estimations, based on groups of the estimations and certainty thereof for the plurality of first measurement point subsets, calculated in the fourth step; a sixth step of statistically calculating estimations of the predetermined number of parameters and certainty of the estimations for each of the plurality of second measurement point subsets, based on measurement results of the fourth number of position-measurement-points; and a seventh step of comparing the statistically valid estimations and certainty with the estimations and certainty for each of the plurality of second measurement point subsets and evaluating possibility of adopting one of the plurality of second measurement point subsets as a measurement point set used to calculate the predetermined number of parameters.
In this case, the statistically valid estimations and their certainty of the predetermined number of parameters calculated based on the position measurement results at the position-measurement-points of the plurality of first measurement point subsets are compared with the estimations and their certainty of the predetermined number of parameters calculated based on the position measurement results at the position-measurement-points of each second measurement point subset. In this comparison, the certainties of the two groups of the estimations are compared as well as the groups of the estimations, the certainties each reflecting deviation of the position error distribution of position-measurement-points of the respective measurement point subset. And by examining the two comparison results, the two position error distributions are compared. Therefore, it can be determined whether or not one of the plurality of second measurement point subsets and the first measurement point subset equally reflect the entire set of all position-measurement-points.
Furthermore, in the fourth step, certainties of position measurement results of the position-measurement-points can be taken in account for calculating the estimations and certainty thereof. In this case, since the estimations and their certainty of the predetermined number of parameters are calculated considering the certainties of the position measurement results at the position-measurement-points, statistically more valid values of the predetermined number of parameters can be calculated.
Furthermore, the fourth step can comprise an estimation step of calculating respective positions of the third number of position-measurement-points, for each of the plurality of first measurement point subsets, based on measurement results of the third number of position-measurement-points and estimating probability density functions that each represent occurrence probability of the calculated position of a respective point of the third number of position-measurement-points; a probability density calculation step of calculating respective probability densities of the calculated positions of the position-measurement-points, based on the probability density functions; and a parameter calculation step of evaluating an error of each of the calculated positions relative to respective reference position using the respective calculated probability density""s value as a piece of weight information and calculating estimations of the predetermined number of parameters, based on the evaluated errors. In this case, since, for each of the first measurement point subsets, errors between the calculated positions and their reference positions are weighted in accordance with the information of the certainties of the calculated positions of the position-measurement-points, i.e. the probability densities at the calculated positions of the position-measurement-points, statistically valid estimations of the predetermined number of parameters can be calculated which uniquely specify position information of any area on an object and rationally reflect the certainties of the calculated positions of the position-measurement-points.
In addition, in the sixth step, certainties of position measurement results of the position-measurement-points can be taken into account upon calculating the estimations and certainty thereof. And the sixth step can comprise an estimation step of calculating respective positions of the fourth number of position-measurement-points, for each of the plurality of second measurement point subsets, based on measurement results of the fourth number of position-measurement-points and estimating probability density functions that each represent occurrence probability of the calculated position of a respective point of the fourth number of position-measurement-points; a probability density calculation step of calculating respective probability densities of the calculated positions of the position-measurement-points, based on the probability density functions; and a parameter calculation step of evaluating an error of each of the calculated positions relative to respective reference position using the respective calculated probability density""s value as a piece of weight information and calculating estimations of the predetermined number of parameters, based on the evaluated errors.
Furthermore, the fourth position detection method can further comprise the eighth step which, if the third step finds second measurement point subsets that can replace the first measurement point subset, selects the most valid, second measurement point subset for replacement and adopts estimations of the predetermined number of parameters, calculated based on measurement results of the fourth number of position-measurement-points of the selected second measurement point subset, as values thereof, and which, if the third step finds no second measurement point subsets that can replace the first measurement point subset, adopts the statistically valid estimations as values of the predetermined number of parameters. Therefore, the number of position-measurement-points used to calculate the values of the predetermined number of parameters can be reduced while maintaining the statistical validity, and improvement of the position detection speed can be achieved maintaining the accuracy.
In addition, in the fourth position detection method, the third step can comprise a fourth step of statistically calculating, for each of the second measurement point subsets, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of the fourth number of position-measurement-points; a fifth step of calculating position errors of all the position-measurement-points of the plurality of first measurement point subsets through use of the estimations of the predetermined number of parameters calculated for each of the second measurement point subsets and evaluating possibility of replacing the plurality of first measurement point subsets by one of the plurality of second measurement point subsets.
In this case, by calculating position errors of position-measurement-points of the plurality of first measurement point subsets by using the estimations, of the predetermined number of parameters, calculated based on the position measurement results at the position-measurement-points of each second measurement point subset, the position error distribution for all position-measurement-points, which will be estimated if the plurality of first measurement point subsets serve as the sample set, can be obtained. Therefore, without calculating groups of the estimations and their certainty of the predetermined number of parameters on the basis of the position measurement results at the position-measurement-points of the plurality of first measurement point subsets and thus the statistically valid estimations and their certainty of the predetermined number of parameters, it can be determined whether or not one of the plurality of second measurement point subsets reflects the-entire set of all position-measurement-points.
Furthermore, in the fourth step, certainties of position measurement results of the position-measurement-points can be taken in account for calculating the estimations and certainty thereof. In this case, since the estimations and their certainty of the predetermined number of parameters are calculated considering the certainties of the position measurement results at the position-measurement-points, statistically more valid values of the predetermined number of parameters can be calculated.
Furthermore, the fourth step can comprise an estimation step of calculating respective positions of the fourth number of position-measurement-points, for each of the plurality of second measurement point subsets, based on measurement results of the fourth number of position-measurement-points and estimating probability density functions that each represent occurrence probability of the calculated position of a respective point of the fourth number of position measurement-points; a probability density calculation step of calculating respective probability densities of the calculated positions of the position-measurement-points, based on the probability density functions; and a parameter calculation step of evaluating an error of each of the calculated positions relative to respective reference position using the respective calculated probability density""s value as a piece of weight information and calculating estimations of the predetermined number of parameters, based on the evaluated errors. In this case, since, for each of the second measurement point subsets, errors between the calculated positions and their reference positions are weighted in accordance with the information of the certainties of the calculated positions of the position-measurement-points, i.e. the probability densities at the calculated positions of the position-measurement-points, statistically valid estimations of the predetermined number of parameters can be calculated which uniquely specify position information of any area on an object and rationally reflect the certainties of the calculated positions of the position-measurement-points.
In addition, the fourth position detection method according to this invention can further comprise the fourth step which, if the third step finds second measurement point subsets that can replace the first measurement point subset, selects the most valid one, for replacement, of the second measurement point subsets and adopts estimations of the predetermined number of parameters, calculated based on measurement results of the fourth number of position-measurement-points of the selected second measurement point subset, as values thereof, and which, if the third step finds no second measurement point subsets that can replace the first measurement point subset, statistically calculates estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of the third number of position-measurement-points for each of the plurality of first measurement point subsets; and adopts as values of the predetermined number of parameters statistically valid estimations thereof calculated based on groups of the estimations and certainty thereof for the plurality of first measurement point subsets. Therefore, the number of position-measurement-points used to calculate the values of the predetermined number of parameters can be reduced while maintaining the statistical validity, and improvement of the position detection speed can be achieved maintaining the accuracy.
Note that in the fourth position detection method of this invention, in the same manner as in the second position detection method, the statistically valid value of each of the predetermined number of parameters can be obtained by calculating average of the corresponding estimations weighted with the respective certainties, each of the certainties representing a piece of weight information for the respective estimation.
Furthermore, in the fourth position detection method of this invention, position measurement marks can be formed at the position-measurement-points in the same manner as in the first position detection method. And a plurality of divided areas each of which is provided with the position measurement marks can be arranged on the object.
According to the fifth aspect of this invention, there is provided a first position detector that detects position information of any area on an object provided with a plurality of position-measurement-points, the position detector comprising a measurement unit that measures pieces of position information of more position-measurement-points than a minimum number of measurements required to calculate values of a predetermined number of parameters, which uniquely specify position information of any area on the object, the position-measurement-points being selected from the plurality of position-measurement-points; an estimation unit that is electrically connected to the measurement unit and that detects respective positions of the selected position-measurement-points, based on the measurement results of the pieces of position information, estimates probability density functions which each represent occurrence probability of the detected position for respective one of the selected position-measurement-points, and calculates probability density of the detected position of each of the position-measurement-points; and a parameter calculation unit that is electrically connected to the measurement unit and the estimation unit and that evaluates detection error of each of the detected positions while using the respective calculated probability density""s value as a piece of weight information and calculates such values of the predetermined number of parameters that the detection errors become statistically minimum as a whole, based on the detection errors.
In this detector, according to the first position detection method of the present invention, the estimation unit calculates the positions of the selected marks (position-measurement-points) and respective probability densities at the calculated mark positions on the basis of position information of the marks measured by the measurement unit. And the parameter calculation unit calculates the values of the predetermined number of parameters that uniquely specify position information of any area on an object. Therefore, the predetermined number of parameters can be accurately calculated, and the position information of any area on an object can be accurately detected.
In the first position detector of the present invention, the measurement unit can comprise an image pickup unit for picking up images of marks formed on the object. In this case, the position information of a selected mark can be measured on the basis of changes in light intensity according to position in the picked-up mark image.
According to the sixth aspect, there is provided a second position detector that detects position information of any area on an object provided with a first number of position-measurement-points, the position detector comprising a measurement unit that measures positions of the position-measurement-points; a set-selection unit that selects a plurality of measurement point subsets which each consist of a third number of position-measurement-points and are different from one another, the third number being larger than a second number and smaller than the first number, the second number being a minimum number of measurement points required to calculate a predetermined number of parameters that uniquely specify position information of any area on the object; and an estimation computing unit that is electrically connected to the measurement unit and the set-selection unit and that statistically calculates, for each of the plurality of measurement point subsets, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of the third number of position-measurement-points.
In this detector, for each measurement point subset selected by the set selection unit, according to the second position detection method of the present invention, the estimation computing unit statistically estimates values of the predetermined number of parameters which uniquely specify position information of any area on an object and calculates the certainty of the estimations on the basis of the positions of position-measurement-points measured by the measurement unit. Therefore, the position distribution of all position-measurement-points can be accurately estimated based on respective groups of the estimations and their certainty of the predetermined number of parameters for the plurality of measurement point subsets that are selected empirically or arbitrarily.
In the second position detector according to this invention, the estimation computing unit can comprise an estimation unit that, for each of the plurality of measurement point subsets, detects respective positions of the third number of position-measurement-points, based on the measurement results of the pieces of position information of the third number of position-measurement-points, estimates probability density functions which each represent occurrence probability of the detected position for the respective point of the third number of position-measurement-points, and calculates probability density of the detected position of each of the position-measurement-points; and a parameter calculation unit that is electrically connected to the estimation unit and that evaluates detection error of each of the detected positions while using the respective calculated probability density""s value as a piece of weight information and calculates such values of the predetermined number of parameters that the detection errors become statistically minimum as a whole, based on the detection errors. In this case, for each of the plurality of measurement point subset, the parameter calculation unit weights errors between the calculated positions and their reference positions in accordance with the information of the certainties of the calculated positions of the position-measurement-points calculated by the estimation unit, i.e. the probability densities at the calculated positions of the position-measurement-points, and calculates estimations of the predetermined number of parameters which uniquely specify position information of any area on an object and rationally reflect the certainties of the calculated positions of the position-measurement-points. Therefore, statistically valid estimations of the predetermined number of parameters can be obtained.
In addition, the second position detector according to this invention can further comprise a parameter value determining unit that is electrically connected to the estimation computing unit and that calculates statistically valid estimations of the predetermined number of parameters based on groups of the estimations and certainty thereof, calculated by the estimation computing unit, for the plurality of measurement point subsets. In this case, the statistically valid values of the predetermined number of parameters for all position-measurement-points can be accurately obtained.
According to the seventh aspect of this invention, there is provided a third position detector that detects position information of any area on an object provided with a first number of position-measurement-points, the position detector comprising a measurement unit that measures positions of the position-measurement-points; a set-selection unit that selects a first measurement point subsets, which each consist of a third number of position-measurement-points, and a plurality of second measurement point subsets which each consist of a fourth number of position-measurement-points and are different from one another, the third number being larger than a second number and smaller than the first number, the second number being a minimum number of measurement points required to calculate a predetermined number of parameters that uniquely specify position information of any area on the object, the fourth number being larger than the second number and smaller than the third number; and an evaluation computing unit that is electrically connected to the set-selection unit and that evaluates possibility of replacing the first measurement point subset by one of the plurality of second measurement point subsets, the first measurement point subset being used to calculate the predetermined number of parameters.
In this detector, according to the third position detection method of this invention, the evaluation computing unit statistically evaluates based on positions of position-measurement-points measured by the measurement unit whether or not it is possible to replace the first measurement point subset as a sample set composed of position-measurement-points to be measured to calculate values of the predetermined number of parameters by one of the plurality of second measurement point subsets each of which is composed of a fewer number of elements than the first measurement point subset. Therefore, upon reducing the number of position-measurement-points as elements of a sample set, statistical validity of the calculated values of the predetermined number of parameters can be maintained.
In the position detector according to this invention, the evaluation computing unit can comprise an estimation calculation unit that is electrically connected to the measurement unit and that statistically calculates, for the specific measurement point subset, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of position information of position-measurement-points composing the specific measurement point subset which is selected from the first measurement point subset and the plurality of second measurement point subsets; and an evaluation unit that is electrically connected to the estimation calculation unit and that compares the estimations and certainty of the first measurement point subset with the estimations and certainty for each of the plurality of second measurement point subsets and evaluates possibility of replacing the first measurement point subset by one of the plurality of second measurement point subsets, the first measurement point subset being used to calculate the predetermined number of parameters.
In this case, the evaluation unit compares the estimations and their certainty of the predetermined number of parameters calculated by the estimation computing unit for the first measurement point subset with those calculated for each second measurement point subset. In this comparison, the certainties of respective groups of the estimations of the two measurement point subsets are compared as well as the groups of the estimations, the certainties each reflecting deviation of the position error distribution of position-measurement-points of the respective measurement point subset. And by examining the two comparison results, the position error distribution of position-measurement-points of the first measurement point subset is compared with that of position-measurement-points of the second measurement point subset. Therefore, it can be determined whether or not one of the plurality of second measurement point subsets and the first measurement point subset equally reflect the entire set of all position-measurement-points.
Furthermore, the estimation calculation unit can comprise an estimation unit that detects respective positions of position-measurement-points composing the specific measurement point subset, based on the measurement results of position information of position-measurement-points composing the specific measurement point subset, estimates probability density functions which each represent occurrence probability of the detected position for respective one of the position-measurement-points of the specific measurement point subset, and calculates probability density of the detected position of each of the position-measurement-points; and a parameter calculation unit that is electrically connected to the estimation unit and that evaluates detection error of each of the detected positions while using the respective calculated probability density""s value as a piece of weight information and calculates such values of the predetermined number of parameters that the detection errors become statistically minimum as a whole, based on the detection errors. In this case, since, for the specific measurement point subset, the parameter calculation unit weights errors between the calculated positions and their reference positions in accordance with the information of the certainties of the calculated positions of the position-measurement-points, i.e. the probability densities at the calculated positions of the position-measurement-points, and calculates statistically valid estimations of the predetermined number of parameters which uniquely specify position information of any area on an object. Therefore, statistically valid estimations of the predetermined number of parameters that rationally reflect the certainties of the calculated positions of the position-measurement-points can be obtained.
In the third position detector according to this invention, the evaluation computing unit can comprise an estimation calculation unit that is electrically connected to the measurement unit and that statistically calculates, for the specific measurement point subset, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of position information of position-measurement-points composing the specific measurement point subset, which is selected from the plurality of second measurement point subsets; an evaluation unit that is electrically connected to the estimation calculation unit and that calculates position errors of the position-measurement-points, composing the first measurement point subset, through use of estimations of the predetermined number of parameters for each of the polarity of second measurement point subsets and evaluates possibility of replacing the first measurement point subset by one of the plurality of second measurement point subsets.
In this case, by the evaluation unit calculating position errors of position-measurement-points composing in the first measurement point subset by using the estimations, calculated by the estimation computing unit, of the predetermined number of parameters for the second measurement point subset, the position error distribution of position-measurement-points in the first measurement point subset can be obtained. Therefore, without calculating the estimations and their certainty of the predetermined number of parameters on the basis of the position measurement results at the position-measurement-points of the first measurement point subset, it can be determined whether or not one of the plurality of second measurement point subsets and the first measurement point subset equally reflect the entire set of all position-measurement-points.
In addition, the estimation calculation unit can comprise an estimation unit that detects respective positions of position-measurement-points composing the specific measurement point subset, based on the measurement results of position information of position-measurement-points composing the specific measurement point subset, estimates probability density functions which each represent occurrence probability of the detected position for respective one of the position-measurement-points of the specific measurement point subset, and calculates probability density of the detected position of each of the position-measurement-points; and a parameter calculation unit that is electrically connected to the estimation unit and that evaluates detection error of each of the detected positions while using the respective calculated probability density""s value as a piece of weight information and calculates such values of the predetermined number of parameters that the detection errors become statistically minimum as a whole, based on the detection errors. In this case, since, for the specific measurement point subset, the parameter calculation unit weights errors between the calculated positions and their reference positions in accordance with the information of the certainties of the calculated positions of the position-measurement-points, i.e. the probability densities at the calculated positions of the position-measurement-points, and calculates statistically valid estimations of the predetermined number of parameters which uniquely specify position information of any area on an object. Therefore, statistically valid estimations of the predetermined number of parameters that rationally reflect the certainties of the calculated positions of the position-measurement-points can be obtained.
Furthermore, the third position detector according to this invention can further comprise a parameter value determining unit that is electrically connected to the evaluation computing unit and that calculates values of the predetermined number of parameters, based on evaluation results of the evaluation computing unit. Therefore, the number of position-measurement-points used to calculate the values of the predetermined number of parameters can be reduced while maintaining the statistical validity, and improvement of the position detection speed can be achieved maintaining the accuracy.
According to the eighth aspect of this invention, there is provided a fourth position detector that detects position information of any area on an object provided with a first number of position-measurement-points, the position detector comprising a measurement unit that measures positions of the position-measurement-points; a set-selection unit that selects a plurality of first measurement point subsets, which each consist of a third number of position-measurement-points and are different from one another, and a plurality of second measurement point subsets which each consist of a fourth number of position-measurement-points and are different from one another, the third number being larger than a second number and smaller than the first number, the second number being a minimum number of measurement points required to calculate a predetermined number of parameters that uniquely specify position information of any area on the object, the fourth number being larger than the second number and smaller than the third number; and an evaluation computing unit that is electrically connected to the set-selection unit and that evaluates possibility of adopting one of the plurality of second measurement point subsets as a measurement point subset to calculate the predetermined number of parameters.
In this detector, according to the fourth position detection method of this invention, the evaluation computing unit statistically evaluates based on positions of position-measurement-points measured by the measurement unit whether or not it is possible to replace the plurality of first measurement point subset as a sample set composed of position-measurement-points to be measured to calculate values of the predetermined number of parameters by one of the plurality of second measurement point subsets each of which is composed of a fewer number of elements than the first measurement point subset. Therefore, upon reducing the number of position-measurement-points as elements of a sample set, statistical validity of the calculated values of the predetermined number of parameters can be maintained.
In the fourth position detector according to this invention, the evaluation computing unit can comprise an estimation calculation unit that is electrically connected to the measurement unit and that statistically calculates, for the specific measurement point subset, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of position information of position-measurement-points composing the specific measurement point subset which is selected from the plurality of first measurement point subset and the plurality of second measurement point subsets, and calculates statistically valid estimations of the predetermined number of parameters and certainty of the estimations, based on groups of estimations of the predetermined number of parameters and certainty of the estimations for the plurality of first measurement point subsets; an evaluation computing unit that is electrically connected to the estimation calculation unit and that compares the statistically valid estimations and certainty of the first measurement point subset with the estimations and certainty for each of the plurality of second measurement point subsets, and evaluates possibility of adopting one of the plurality of second measurement point subsets as a measurement point subset to calculate the predetermined number of parameters.
In this case, the evaluation unit calculates the statistically valid estimations and their certainty of the predetermined number of parameters based on sets of the predetermined number of parameters, for the plurality of first measurement point subsets, calculated by the estimation calculation unit, and compares the statistically valid estimations and their certainty of the predetermined number of parameters, for each second measurement point subset, calculated by the estimation calculation unit with the statistically valid estimations and their certainty of the predetermined number of parameters. In this comparison, the certainties of the two groups of the estimations are compared as well as the groups of the estimations, the certainties each reflecting deviation of the position error distribution of position-measurement-points of the respective measurement point subset. And by examining the two comparison results, the two position error distributions are compared. Therefore, it can be determined whether or not one of the plurality of second measurement point subsets and the first measurement point subset equally reflect the entire set of all position-measurement-points.
Furthermore, the estimation calculation unit can comprise an estimation unit that is electrically connected to the measurement unit and that detects respective positions of position-measurement-points composing the specific measurement point subset, based on the measurement results of position information of position-measurement-points composing the specific measurement point subset, estimates probability density functions which each represent occurrence probability of the detected position for respective one of the position-measurement-points of the specific measurement point subset, and calculates probability density of the detected position of each of the position-measurement-points; and a parameter calculation unit that is electrically connected to the estimation unit and that evaluates detection error of each of the detected positions while using the respective calculated probability density""s value as a piece of weight information and calculates such values of the predetermined number of parameters that the detection errors become statistically minimum as a whole, based on the detection errors. In this case, since, for the specific measurement point subset, the parameter calculation unit weights errors between the calculated positions and their reference positions in accordance with the information of the certainties of the calculated positions of the position-measurement-points, i.e. the probability densities at the calculated positions of the position-measurement-points, and calculates statistically valid estimations of the predetermined number of parameters which uniquely specify position information of any area on an object. Therefore, statistically valid estimations of the predetermined number of parameters that rationally reflect the certainties of the calculated positions of the position-measurement-points can be obtained.
In the fourth position detector according to this invention, the evaluation computing unit can comprise an estimation calculation unit that is electrically connected to the measurement unit and that statistically calculates, for the specific measurement point subset, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of position information of position-measurement-points composing the specific measurement point subset which is selected from the plurality of second measurement point subsets; and an evaluation unit that is electrically connected to the estimation calculation unit and that calculates errors of all the position-measurement-points of the plurality of first measurement point subsets through use of the estimations of the predetermined number of parameters-calculated for each of the second measurement point subsets, and evaluates possibility of replacing the plurality of first measurement point subsets by one of the plurality of second measurement point subsets.
In this case, by the evaluation unit calculating position errors of position-measurement-points of the plurality of first measurement point subsets by using the estimations, of the predetermined number of parameters for each second measurement point subset, calculated by the estimation calculation unit, the position error distribution for all position-measurement-points, which will be estimated if the plurality of first measurement point subsets serve as the sample set, can be obtained. Therefore, without calculating groups of the estimations and their certainty of the predetermined number of parameters on the basis of the position measurement results at the position-measurement-points of the plurality of first measurement point subsets and thus the statistically valid estimations and their certainty of the predetermined number of parameters, it can be determined whether or not one of the plurality of second measurement point subsets reflects the entire set of all position-measurement-points.
In addition, the estimation calculation unit can comprise an estimation unit that detects respective positions of position-measurement-points composing the specific measurement point subset, based on the measurement results of position information of position-measurement-points composing the specific measurement point subset, estimates probability density functions which each represent occurrence probability of the detected position for respective one of the position-measurement-points of the specific measurement point subset, and calculates probability density of the detected position of each of the position-measurement-points; and a parameter calculation unit that is electrically connected to the estimation unit and that evaluates detection error of each of the detected positions while using the respective calculated probability density""s value as apiece of weight information and calculates such values of the predetermined number of parameters that the detection errors become statistically minimum as a whole, based on the detection errors. In this case, since, for the specific measurement point subset, the parameter calculation unit weights errors between the calculated positions and their reference positions in accordance with the information of the certainties of the calculated positions of the position-measurement-points, i.e. the probability densities at the calculated positions of the position-measurement-points, and calculates statistically valid estimations of the predetermined number of parameters which uniquely specify position information of any area on an object. Therefore, statistically valid estimations of the predetermined number of parameters that rationally reflect the certainties of the calculated positions of the position-measurement-points can be obtained.
The fourth position detector according to this invention can further comprise a parameter value determining unit that is electrically connected to the parameter calculation unit and that calculates values of the predetermined number of parameters, based on evaluation results of the evaluation computing unit. Therefore, the number of position-measurement-points used to calculate the values of the predetermined number of parameters can be reduced while maintaining the statistical validity, and improvement of the position detection speed can be achieved maintaining the accuracy.
According to the ninth aspect of this invention, there is provided an exposure method for transferring a predetermined pattern onto divided areas on a substrate, comprising an arrangement information calculation step of calculating a predetermined number of parameters that pertain to positions of the divided areas by a position detection method according to this invention and calculating arrangement information of the divided areas on the substrate; and a transfer step of transferring the pattern onto the divided areas while aligning the substrate based on the arrangement information of the divided areas calculated in the arrangement information calculation step.
According to this method, a pattern is transferred onto divided areas while accurately detecting the arrangement of the divided areas on a substrate using a detection method of the present invention and aligning the substrate on the basis of the detection results. Therefore, a pattern can be accurately transferred onto the divided areas.
According to the tenth aspect of this invention, there is provided an exposure apparatus that transfers a predetermined pattern onto divided areas on a substrate, comprising a stage unit that moves the substrate along a movement plane; and a position detector according to this invention that calculates arrangement information of the divided areas on the substrate mounted on the stage unit.
This apparatus transfers a pattern onto divided areas while moving and aligning the substrate through the stage unit on the basis of arrangement of the divided areas detected by the position detection unit of this invention. Therefore, a pattern can be accurately transferred onto the divided areas.
In addition, in a lithography process, by performing exposure using an exposure apparatus according to the present invention, it is possible to form a multi-layer pattern on a substrate with high accuracy of superposition, and therefore it is possible to manufacture a more highly integrated micro device with high yield, and the productivity can be improved. Accordingly, another aspect of the present invention is a device manufactured by using an exposure apparatus of the present invention and a method of manufacturing a device using an exposure method of the present invention.